• Previous Article
    Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy
  • EECT Home
  • This Issue
  • Next Article
    Periodic solutions and multiharmonic expansions for the Westervelt equation
June  2021, 10(2): 249-258. doi: 10.3934/eect.2020064

Decay rate of global solutions to three dimensional generalized MHD system

1. 

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

2. 

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

* Corresponding author: Yinxia Wang

Received  September 2019 Published  June 2020

Fund Project: The second author is supported in part by the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province(Grant No. 20ZX002)

We investigate the initial value problem for the three dimensional generalized incompressible MHD system. Analyticity of global solutions was proved by energy method in the Fourier space and continuous argument. Then decay rate of global small solutions in the function space $ \mathcal {X}^{1-2\alpha}\bigcap \mathcal {X}^{1-2\beta} $ follows by constructing time weighted energy inequality.

Citation: Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.  Google Scholar

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1. Google Scholar

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[6]

J. FanF. LiG. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.  Google Scholar

[7]

C. HeX. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.  Google Scholar

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.  Google Scholar

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.  Google Scholar

[12]

Y. LinH. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.  Google Scholar

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.  Google Scholar

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.  Google Scholar

[16]

W. WangT. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.  Google Scholar

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.  Google Scholar

[20]

Y. XiaoB. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.  Google Scholar

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.  Google Scholar

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1. Google Scholar

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.  Google Scholar

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar

show all references

References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.  Google Scholar

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1. Google Scholar

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[6]

J. FanF. LiG. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.  Google Scholar

[7]

C. HeX. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.  Google Scholar

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.  Google Scholar

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.  Google Scholar

[12]

Y. LinH. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.  Google Scholar

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.  Google Scholar

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.  Google Scholar

[16]

W. WangT. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.  Google Scholar

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.  Google Scholar

[20]

Y. XiaoB. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.  Google Scholar

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.  Google Scholar

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1. Google Scholar

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.  Google Scholar

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar

[1]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198

[2]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210

[3]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[4]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[5]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[6]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[7]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007

[8]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[9]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[10]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[11]

Ethan Akin, Julia Saccamano. Generalized intransitive dice II: Partition constructions. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021005

[12]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[13]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[14]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[15]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[16]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[17]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[18]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[19]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[20]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (79)
  • HTML views (284)
  • Cited by (0)

Other articles
by authors

[Back to Top]